TAM3B DIFFERENTIAL EQUATIONS Unit : I to V
|
|
- Amanda Pearson
- 5 years ago
- Views:
Transcription
1 TAM3B DIFFERENTIAL EQUATIONS Unit : I to V
2 Unit I -Syllabus Homogeneous Functions and examples Homogeneous Differential Equations Exact Equations First Order Linear Differential Equations Reduction of order TAM3B-Differential Equations 2
3 Homogeneous Function A functionf(x,y)iscalledhomogeneous n of degree nif f(tx,ty) = t f(x,y) Example1: 2 f(x,y) = x + xy isahomogeneousfunctionof degree2 Example2 : f(x,y) = sin(x / y) isahomogeneousfunctionof degree0 TAM3B-Differential Equations 3
4 Homogeneous Differential Equations ThedifferentialequationM( x,y) dx +N( x,y) dy = 0 issaidtobehomogeneous if MandNarehomogeneousfunctions of samedegree Theequationcanbe writteninthe form dy = f ( x,y ) dx -M x,y where f ( x,y ) =,isahomogeneousof degree0 N x,y ( ) ( ) TAM3B-Differential Equations 4
5 Procedure for Solving Homogeneous Equation dy Step1:Rewrite thegivenequationinthe form = dx dy dz Step2:Put y = zx and = z + x dx dx Step3:Seperate the variableszandx Step 4:Integrateandget zandx y Step5:Replace zby x ( ) ( ) -M x,y N x,y TAM3B-Differential Equations 5
6 Illustration Problemsforpractice Problem1:Solve(x + y)dx -(x - y)dy = Problem2:Solve(x - 2y )dx + xydy = 0 Problem3:Solvexy = 2x +3y Problem4:Solvex 2 y = y 2 + 2xy TAM3B-Differential Equations 6
7 Exact Equation M The expressionmdx + Ndyissaidtobeexact if = y Example: 2 x + dy ydx 0 y + = 2 M N M= yn, = x+ = 1, = 1 y y x M N = Theequationisexact y x N x TAM3B-Differential Equations 7
8 Solving an Exact Equation Step 1: Check whether the given equation is exact or not. If exact proceed to step 2 Step2: I1 = M ( x, y) dx, treatingyasconstant Step3: I2 = N( x, y) dy, Neglectingthe'x'terms Step4:Solutionis I1+ I2 = c TAM3B-Differential Equations 8
9 Illustration Problemsforpractice Problem1:Test theequationfor exactnessand solve it if it is exact e dx + ( xe + 2 y) dy = 0 y Problem2:Test theequationfor exactnessand solve it if it is exact ( y + y cos xy) dx + ( x + xcos xy) dy = 0 Problem 3:Test theequationfor exactnessand 2 solve it if it is exact (2y 4x + 5) dx (4 2y + 4 xy) dy = 0 y TAM3B-Differential Equations 9
10 Solving First Order Linear Equation The general first order linear equation is dy Pxy ( ) Qx ( ) dx + = Step1:Integrating Factor (. IF) = e Pdx Step2:Find Q(. I F) dx Step 3: Solution is yif (. ) = QIF (. ) dx+ c TAM3B-Differential Equations 10
11 Illustration Problems for practice dy Problem1: Solve x 3y = x dx 4 Problem2: Solve (2 y x ) dx = xdy 3 Problem3: Solve 1 y + y = 1 + e 2x Problem4: Solve (1 + x ) dy + 2xydx = cot xdx 2 TAM3B-Differential Equations 11
12 Reduction of Order The general second order differential equation has the form Fxyy (,,, y ) = 0 There are two special types of second order equations that can be solved by first order methods. Type 1: Dependent variable missing Type 2: Independent variable missing TAM3B-Differential Equations 12
13 Dependent Variable Missing If y is not present then the general second order Differential equation can be written as f( xy,, y ) = 0 dp Step1:Put y = p and y = dx Step2:Substitute in the given equation and seperate the variables Step3:Integrate and get the value of p in terms of x dy Step4:Replace pby and seperate the variables yand x dx Step 5:Integrate to get the solution. TAM3B-Differential Equations 13
14 Independent Variable Missing If x is not present then the general second order Differential equation can be written as gyy (,, y ) = 0 dp Step1:Put y = p and y = p dy Step2:Substitute in the given equation and seperate the variables Step3:Integrate and get the value of p in terms of y dy Step4:Replace pby and seperate the variables yand x dx Step 5:Integrate to get the solution. TAM3B-Differential Equations 14
15 Problems for practice 2 Problem1: Solve yy + ( y ) = 0 Problem2: Solve xy = y + ( y ) 3 2 = Problem3: Solve y k y 0 Problem4: Solve xy + y = 4x TAM3B-Differential Equations 15
16 Problems for practice Problem5: Solve xy = y + 2xe y x Problem6: Check for exactness and solve if it is exact 2 cos xcos ydx + 2sin xsin y cos ydy = 0 Problem7: Solve xy = 2 xy + ( y ) 2 Problem8: Solve xy + y = x y 4 3 TAM3B-Differential Equations 16
17 Unit II- Syllabus Second order linear differential equations Types of Solutions Complementary Function and its types Particular Integral and its types Variation of Parameters TAM3B-Differential Equations 17
18 Second order linear Differential Equations The general linear differential equation of order n is of the form n dy n-1 d y n 1 n-1 n +a +...+a y = f(x), dx dx where are a 1,a 2,...,a n are real constant. This equation can also be written in operator form as n n-1 ( 1 n) ( ) D +a D +...+a y = f x (1) TAM3B-Differential Equations 18
19 Types of Solution The solution consists of two parts (i) Complementary funtion (ii) Particular integral. i.e., y = y + y where y is a complementary function, c p c y is particular integral p TAM3B-Differential Equations 19
20 Complementary Function To find complementary function we have to form the auxillary equation which is obtained by putting D = m and f(x)=0. Thus the auxillary equation of (1) is given by m am am a n n n 1 n = 0 (2) Equation (2) is an ordinary algebraic equation in m of degree 'n'. By solving this equation we get n roots for m, say m, m,..., m 1 2 n TAM3B-Differential Equations 20
21 Types of Complementary Functions Type(i): If all the roots m,m,...,m are real and different then the 1 2 n mx 1 m2x mx 3 complementary function (C.F) y = Ae +Be +Ce +... Type(ii): If any two roots are equal say m = m = m then the complementary fu ( ) c 1 2 mx nction (C.F) y = Ax +B e Type(iii): If the roots are imaginary say m = α ± iβ the the complementary function (C.F) y = e c c αx ( Acosβx +Bsinβx) TAM3B-Differential Equations 21
22 Particular Integral When the R.H.S of the given differential equation is zero, we need not find particular integral. In otherwords the complementary function gives the complete solution. When R.H.S of a given differential equation is a function of ( ) ( ) ax ax x say e,sinax or cosax, Algebraic function, e f x,xf x, we have to find particular integral. TAM3B-Differential Equations 22
23 Type 1 ax If f(x) = e, then the particular integral is given by ( ) 1 P.I = f D ( ) 1 f a ( ) ( a) ax ( ) ax = e, provided f a 0 1 f ( ) ax If f a = 0, then P.I = x. e, provided f a ax If f ( a ) = 0, then P.I = x. e, provided f ( a) 0 f a e ( ) TAM3B-Differential Equations 23
24 Type 2 1 If f ( x ) = sinax or cosax, then P.Iis given by P.I = f D 2 2 ( ) ( ) In f D replace D by - a,provided f D 0. ( ) ( ) ( ) 1 f ( D) ( D) ( ) ( ) ( ) 2 2 If f D = 0, then P.I = x. sinax (or) cosax,(d -a ) provided f D 0. 1 f 2 If f D = 0, then P.I = x. si nax (or) cosax sinax (or) cosax 2 2 Again replace D by - a in f D provided f D 0 and so on. TAM3B-Differential Equations 24
25 Type 3 If f(x) = a x +a x n n n ( ) ( ) ( n n-1 ) 0 1 n -1 n n-1 ( ) ( 0 1 n) a where f(x) is a pure algebraic function then 1 P.I = a x +a x +...+a f D = f D a x +a x +...+a Expand f D by using Binomial theorem in ascending powers of D then operete on a x +a x +...+a n n n TAM3B-Differential Equations 25
26 Type 4 ( ) ax If f x = e X where X is some function of x, then 1 P.I = f D ( ) ax.e X 1 f D+a ax = e..x ( ) If X is either sinax or cosax then proceed as Type 2 If X is some algebraic expression then proceed as Type 3 TAM3B-Differential Equations 26
27 Method of Variation of Parameters 2 Consider D y +a1dy +a2y = X, where X is a function of x Let the complementary function C.F = c f +c f where c,c are constants and f,f are functions of x. Then P.I = Pf +Qf fx fx 2 1 where P = - dx and Q = dx ff -f f ff -f f Hence complete solution y = c f +c f +P.I TAM3B-Differential Equations 27
28 Illustration Problems for practice Problem1:Solve y + 3y 10y = 6e 4x Problem2:Solve y 2y = 12x 10 Problem3:Solve y 3y + 2y = 14sin 2x 18cos 2x Problem4: Solve y + 4y = tan 2x TAM3B-Differential Equations 28
29 Problems for practice Problem :Solve y + a y = bx 2 5 cos Problem 6: Solve 2 2x ( 13 12) 5 D D+ y = e + e x Problem 7: Solve 2 2 ( 5 6) 2 D D+ y = x x+ Problem 8: Solve 2 x 2 (4 4 5) cos D + D+ y = e x+ e x Problem 9: Solve y 2y + y = 2x TAM3B-Differential Equations 29
30 Unit III- Syllabus Linear Systems Theorems Relating the solutions of the system Theorems Relating the Wronskian Solution of Homogeneous Linear System With Constant Coefficients TAM3B-Differential Equations 30
31 Linear System The linear system dx = a1() tx+ bty 1() + f1() t dt dy = a2() tx+ b2() ty+ f2() t dt is homogeneous if f () t and f () t are identically zero 1 2 TAM3B-Differential Equations 31
32 Theorem A [ ] [ ] If t is any point of the interval ab,, and if x y are o 0 0 any numbers whatever, then the linear system has one x= xt () and only one solution y = yt () valid throughout ab,, such that xt ( ) = x and yt ( ) = y TAM3B-Differential Equations 32
33 Theorem B dx = a1() tx+ bty 1() dt If the homogeneous system has two solutions dy = a2() tx+ b2() ty dt x= x1() t x= x2() t and on [ ab, ], then y = y1() t y = y2() t x= cx 1 1() t + cx 2 2() t y = cy 1 1() t + cy 2 2() t is also a solution on for any constants c and c [ ab, ] 1 2 TAM3B-Differential Equations 33
34 Theorem C x= x () t x= x () t 1 2 If the two solutions and of the y = y1() t y = y2() t dx = a1() tx+ bty 1() dt homogeneous system have a Wronskian W(t) dy = a2() tx+ b2() ty dt x= cx 1 1() t + cx 2 2() t that does not vanish on [ ab, ] then y = cy 1 1() t + cy 2 2() t is the general solution of the system TAM3B-Differential Equations 34
35 Theorem D If Wt () is the Wronskian of the two solutions x= x1() t x= x2() t and of the homogeneous system y = y1() t y = y2() t dx = a1() tx+ bty 1() dt then W(t) is either identically zero dy = a2() tx+ b2() ty dt or nowhere zero on [ ab, ] TAM3B-Differential Equations 35
36 Theorem E x= x1() t x= x2() t If the two solutions and of the y = y1() t y = y2() t dx = a1() tx+ bty 1() dt homogeneous system are linearly independent dy = a2() tx+ b2() ty dt x= cx 1 1() t + cx 2 2() t on [ ab, ] then is the general solution of the system y = cy 1 1() t + cy 2 2() t on this interval. TAM3B-Differential Equations 36
37 Theorem F x= x () t x= x () t y = y1() t y = y2() t dx = a1() tx+ bty 1() dt homogeneous system are linearly independent dy = a2() tx+ b2() ty dt x= xp () t on [ ab, ] and if is any particular solution of the non homogeneous y = yp () t 1 2 If the two solutions and of the x= cx 1 1() t + cx 2 2() t + xp () t system on this interval then is the y = cy 1 1() t + cy 2 2() t + yp () t general solution of the nonhomogeneous system on [ ab, ] TAM3B-Differential Equations 37
38 Solution of Homogeneous Linear System With Constant Coefficients dx = ax 1 + by 1 dt Consider the system dy = ax 2 + by 2 dt mt Step1: Assume x = Ae and y = Be Step 2: Substituting in (1) we get Ame = a Ae + b Be mt mt mt 1 1 mt mt mt = Bme a Ae b Be mt (1) TAM3B-Differential Equations 38
39 Solution of Homogeneous Linear System With Constant Coefficients mt Step 3:Dividing by e we get the linear algebraic system ( a1 m) A+ bb 1 = 0 aa+ ( b mb ) = 0 Step 4 :Find 2 2 a1 m b1 a b m m a1 b2m ab 1 2 ab 2 1 = 0, to get the quadratic equation ( + ) + ( ) = 0 Step 5: Solve the quadratic equation and find the value of m 1 and m 2 TAM3B-Differential Equations 39
40 Types of Solutions Distinct real roots: The general solution is mt 1 m2t x= c1ae 1 + c2ae 2 mt 1 m2t y = cbe c2be 2 Distinct complex roots: The general solution is at x = e [ c1( A1cos bt A2sin bt) + c2( A1sin bt + A2cos bt)] at y = e [ c1( B1cos bt B2sin bt) + c2( B1sin bt + B2 cos bt)] Equal real roots: The general solution is mt x = c1ae + c2( A1+ A2t ) e mt y = c1be + c2( B1 + B2t) e mt mt TAM3B-Differential Equations 40
41 Illustration Problems for Practice Problem1:Find the Wronskian of dx = x+ y dt Problem 2: Solve dy = 4x 2y dt dx = 3x 4y dt Problem 3: Solve dy = x y dt 3t 2t x= e x= e and y = e y = 2e 3t 2t TAM3B-Differential Equations 41
42 Problems for Practice dx = 3x+ 4y dt Problem 4: Solve dy = 2x+ 3y dt dx = 2x dt Problem 5: Solve dy = 3y dt dx = 5x+ 4y dt Problem 6: Solve dy = x+ y dt TAM3B-Differential Equations 42
43 Unit-IV-Syllabus Partial Differential Equations Formation of PDE by eliminating arbitrary constants and functions Types of Solutions Lagrange Equation TAM3B-Differential Equations 43
44 Partial Differential Equations A partial differential equation is an equation involving a function of two or more variables and some of its partial derivatives. Therefore a partial differential equation contains one dependent variable and more than one independent variable. EXAMPLES: 2 u 1. = 2 x 3 u y u u 2. + = 0 x y ( u - dependent variable; x,y -independent variable) ( u - dependent variable; x,y -independent variable) TAM3B-Differential Equations 44
45 Notations 2 z z z p=, q=, r =, 2 x y x t z = and s = 2 y z xy 2 2 TAM3B-Differential Equations 45
46 Formation of PDE by eliminating arbitrary constants Let f ( x,y,z,a,b ) = 0 (1) be an equaion which contains two arbitrary constants 'a' and 'b'. To eliminate two constants we need atleast three equations. Partially differentiating eqn (1) w.r.t. x and y we get two more equations. From these equations we can eliminate the two constants 'a' and 'b'. Similarly for eliminating three constants we need four equations and so on. TAM3B-Differential Equations 46
47 Formation of PDE by eliminating arbitrary functions Formation of Partial Differential Equations by Elimination of Arbitrary Functions : The elimination of one arbitrary function from a given relation gives a partial differential equation of first order while elimination of two arbitrary function gives a second or higher order partial differential equations TAM3B-Differential Equations 47
48 Illustration Problems for practice Problem1:Form the partial differential equations by eliminating the constants from z = ax + by + ab Problem 2:Form the partial differential equations by eliminating the constants from z = x + a y + b 2 2 ( )( ) Problem 3:Form the partial differential equations by eliminating the arbitrary functions from z = x + y + f ( xy) Problem 4:Form the partial differential equations by eliminating xy the arbitrary functions from z = f( ) z TAM3B-Differential Equations 48
49 Types of Solutions (a) A solution in which the number of arbitrary constants is equal to the number of independent variables is called Complete Integral (or) Complete Solution (b) In Complete integral if we give particular values to the arbitrary constants we get Particular Integral. ( xyzab) ( ) (c) Singular Integral: Let f x,y,z,p,q = 0 be a partial differential equation whose complete integral is φ,,,, = 0, Differentiating partially w.r.t. a and b and then φ φ equate to zero we get = 0 and = 0 a b The eliminant of a and b is called Singular Integral. TAM3B-Differential Equations 49
50 Lagrange Equation The equation of the form Pp +Qq =R is known as Lagrange's equation, where P,Q and R are functions of x, y and z. To solve this equation it is enough to solve the subsidiary equations dx dy dz = = P Q R If the solution of the subsidiary equation is of the form ( ) ( ) u x,y = c and v x,y = c then the solution of the 1 2 ( ) Lagrange's equation is φ u,v = 0 TAM3B-Differential Equations 50
51 Methods for Solving Lagrange Equation To solve the subsidiary equation we have two methods (i) Method of grouping (ii) Method of multipliers. (i)in the method of grouping consider any two ratios dx dy dz of the auxillary equation = =, integrate to P Q R get the value of c and c 1 2 The solution is φ( c, c ) = TAM3B-Differential Equations 51
52 Method of Multipliers If we were not able to group the ratios we will use method of multipliers to find c and c 1 2 Consider the multipliers to be ( l, m, n ) and ( l, m, n ) then the auxillary equation will be dx dy dz l1dx + m1dy + n1dz l2dx + m2dy + n2dz = = = = P Q R lp+ mq+ nr lp+ mq+ nr Choose the multipliers in such a way that the denominator becomes zero in the ratios so that numerator can be equated to 0 to find c and c. Otherwise choose the required number of multipliers to find c and c by method of grouping. TAM3B-Differential Equations 52
53 Illustration Problems for Practice Problem1: Solve px + qy = z Problem 2: Solve y zp + zx q = xy Problem 3: Solve xp + yq = 0 Problem 4: Solve xp+ yq= z TAM3B-Differential Equations 53
54 Problems for Practice Problem 5: Solve ( ) = 2 2 zx y xp yq Problem 6: Solve ( y+ z) p+ ( z+ xq ) = x+ y Problem 7: Solve ( y+ z) p ( x+ zq ) = x y Problem 8: Solve ( ) ( ) ( ) xz y p+ yx z q= zy x Problem 9: Solve ( mz ny) p + ( nx lz) q = ly mx Problem10: Solve ( y z) p+ ( z xq ) = x y TAM3B-Differential Equations 54
55 Unit V-Syllabus Charpit s method Special types of Functions Type I Type II Type III Type IV TAM3B-Differential Equations 55
56 Charpit s Method This is the general method of solving a partial differential equation of first order f( xyz,,, pq, ) = 0 The auxillary equation is dp dq dz dx dy = = = = f + pf f + qf pf + qf f f x z y z p q p q Where f f f f f fx =, fy =, fz =, fp =, fq = x y z p q TAM3B-Differential Equations 56
57 Procedure Step1:Find f, f, f, f f x y z p, q Step 2: Write the auxillary equation dp dq dz dx dy = = = = f + pf f + qf pf + qf f f x z y z p q p q Step 3:Integrate the simplest differential equation Step 4: Solve the integral obtained in step 3 and use the given equation to find pand q Step 5: Substitute in dz = pdx + qdy, Step 6:Integrate to get the solution TAM3B-Differential Equations 57
58 Illustration Problems for Practice Problem1: 2 2 ( ) p + q x = pz Problem 2: 2 6yz 6pxy 3qy + pq = 0 Problem 3: 2 2 ( ) p + q y = qz Problem 4: z 2 = pqxy TAM3B-Differential Equations 58
59 Special Types of First Order Equations Type I:Equaions Involving pand qonly f( pq, ) = 0 Step1:Put p= ain the given equation Step 2:Find qin terms of ' a' Step 3: Substitute p and q in z = pdx + qdy Step 4:Integrate to get the complete integral TAM3B-Differential Equations 59
60 Special Types of First Order Equations Type II:Equations Not Involving the Independent Variables f(, z pq, ) = 0 Step1:Put x + ay = u. Let z = f ( x + ay) be the solution dz dz Step 2: Substiute p= and q= a in the given equation du du Step 3: Seperate the variables zand u Step 4:Integrate and get z Step 5:Put u = x + ay to get the solution TAM3B-Differential Equations 60
61 Special Types of First Order Equations Type III: Seperable Equaions f( xp, ) = f( yq, ) 1 2 Step1: Consider f( xp, ) = f( yq, ) = k 1 2 f ( x, p) = k and f ( y, q) = k 1 2 Step 2:Find pin terms of xand k, and qin terms of yand k Step 3: Substitute p and q in dz = pdx + qdy Step 4:Integrate to get the complete integral TAM3B-Differential Equations 61
62 Special Types of First Order Equations Type IV: Clairaut's Form z = px + qy + f ( p, q) Step1:Put p= aand q= bto get the complete integral z = ax + by + f ( a, b) z z Step 2:To get the singular integral find and equate a b to zero eliminate aand b TAM3B-Differential Equations 62
63 Illustration Problems for Practice Problem1: z = px + qy 2 pq Problem 2: z = px + qy + pq Problem 3: z = px + qy + log pq Problem 4: z = px + qy + p q 2 2 Problem 5: p + q = TAM3B-Differential Equations 63
64 Problems for Practice Problem 6: Solve p + q = 1 Problem 7:Solve pq = 1 Problem 8:Find the complete integral of p(1 + q) = qz Problem 9:Find the complete integral of the PDE pz + q = TAM3B-Differential Equations 64
65 Problems for Practice Problem10:Find the complete integral of the PDE p y(1 + x ) = qx Problem11:Find the complete integral of the PDE 2 2 p q x y + = + Problem12:Find the complete integral of the PDE pq = xy TAM3B-Differential Equations 65
66 Problems for Practice Problem13:Find the complete integral of the PDE p + q = pq Problem14:Find the complete integral of the PDE zpq = p + q Problem15:Find the complete integral of the PDE pq + xy = xq( x + y) TAM3B-Differential Equations 66
Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225
Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225 Dr. Asmaa Al Themairi Assistant Professor a a Department of Mathematical sciences, University of Princess Nourah bint Abdulrahman,
More informationCompatible Systems and Charpit s Method
MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 28 Lecture 5 Compatible Systems Charpit s Method In this lecture, we shall study compatible systems of first-order PDEs the Charpit s method for solving
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationPartial Differential Equations
Partial Differential Equations Lecture Notes Dr. Q. M. Zaigham Zia Assistant Professor Department of Mathematics COMSATS Institute of Information Technology Islamabad, Pakistan ii Contents 1 Lecture 01
More informationdx n a 1(x) dy
HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)
More informationSome Special Types of First-Order PDEs Solving Cauchy s problem for nonlinear PDEs. MA 201: Partial Differential Equations Lecture - 6
MA 201: Partial Differential Equations Lecture - 6 Example Find a general solution of p 2 x +q 2 y = u. (1) Solution. To find a general solution, we proceed as follows: Step 1: (Computing f x, f y, f u,
More informationMA 201: Partial Differential Equations Lecture - 2
MA 201: Partial Differential Equations Lecture - 2 Linear First-Order PDEs For a PDE f(x,y,z,p,q) = 0, a solution of the type F(x,y,z,a,b) = 0 (1) which contains two arbitrary constants a and b is said
More informationCompatible Systems and Charpit s Method Charpit s Method Some Special Types of First-Order PDEs. MA 201: Partial Differential Equations Lecture - 5
Compatible Systems and MA 201: Partial Differential Equations Lecture - 5 Compatible Systems and Definition (Compatible systems of first-order PDEs) A system of two first-order PDEs and f(x,y,u,p,q) 0
More informationDepartment of mathematics MA201 Mathematics III
Department of mathematics MA201 Mathematics III Academic Year 2015-2016 Model Solutions: Quiz-II (Set - B) 1. Obtain the bilinear transformation which maps the points z 0, 1, onto the points w i, 1, i
More informationDepartment of Mathematics. MA 108 Ordinary Differential Equations
Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai 476, INDIA. MA 8 Ordinary Differential Equations Autumn 23 Instructor Santanu Dey Name : Roll No : Syllabus and Course Outline
More information1 First Order Ordinary Differential Equation
1 Ordinary Differential Equation and Partial Differential Equations S. D. MANJAREKAR Department of Mathematics, Loknete Vyankatrao Hiray Mahavidyalaya Panchavati, Nashik (M.S.), India. shrimathematics@gmail.com
More informationEssential Ordinary Differential Equations
MODULE 1: MATHEMATICAL PRELIMINARIES 10 Lecture 2 Essential Ordinary Differential Equations In this lecture, we recall some methods of solving first-order IVP in ODE (separable and linear) and homogeneous
More informationLecture Notes on. Differential Equations. Emre Sermutlu
Lecture Notes on Differential Equations Emre Sermutlu ISBN: Copyright Notice: To my wife Nurten and my daughters İlayda and Alara Contents Preface ix 1 First Order ODE 1 1.1 Definitions.............................
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More information2 Linear Differential Equations General Theory Linear Equations with Constant Coefficients Operator Methods...
MA322 Ordinary Differential Equations Wong Yan Loi 2 Contents First Order Differential Equations 5 Introduction 5 2 Exact Equations, Integrating Factors 8 3 First Order Linear Equations 4 First Order Implicit
More informationDepartment of Mathematics, K.T.H.M. College, Nashik F.Y.B.Sc. Calculus Practical (Academic Year )
F.Y.B.Sc. Calculus Practical (Academic Year 06-7) Practical : Graps of Elementary Functions. a) Grap of y = f(x) mirror image of Grap of y = f(x) about X axis b) Grap of y = f( x) mirror image of Grap
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
More information(1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C.
Midterm 1 33B-1 015 October 1 Find the exact solution of the initial value problem. Indicate the interval of existence. y = x, y( 1) = 0. 1 + y Solution. We observe that the equation is separable, and
More informationIndefinite Integration
Indefinite Integration 1 An antiderivative of a function y = f(x) defined on some interval (a, b) is called any function F(x) whose derivative at any point of this interval is equal to f(x): F'(x) = f(x)
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 25 First order ODE s We will now discuss
More information2. Second-order Linear Ordinary Differential Equations
Advanced Engineering Mathematics 2. Second-order Linear ODEs 1 2. Second-order Linear Ordinary Differential Equations 2.1 Homogeneous linear ODEs 2.2 Homogeneous linear ODEs with constant coefficients
More informationDifferential Equations Class Notes
Differential Equations Class Notes Dan Wysocki Spring 213 Contents 1 Introduction 2 2 Classification of Differential Equations 6 2.1 Linear vs. Non-Linear.................................. 7 2.2 Seperable
More information6 Second Order Linear Differential Equations
6 Second Order Linear Differential Equations A differential equation for an unknown function y = f(x) that depends on a variable x is any equation that ties together functions of x with y and its derivatives.
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
More informationA. MT-03, P Solve: (i) = 0. (ii) A. MT-03, P. 17, Solve : (i) + 4 = 0. (ii) A. MT-03, P. 16, Solve : (i)
Program : M.A./M.Sc. (Mathematics) M.A./M.Sc. (Previous) Paper Code:MT-03 Differential Equations, Calculus of Variations & Special Functions Section C (Long Answers Questions) 1. Solve 2x cos y 2x sin
More informationREFERENCE: CROFT & DAVISON CHAPTER 20 BLOCKS 1-3
IV ORDINARY DIFFERENTIAL EQUATIONS REFERENCE: CROFT & DAVISON CHAPTER 0 BLOCKS 1-3 INTRODUCTION AND TERMINOLOGY INTRODUCTION A differential equation (d.e.) e) is an equation involving an unknown function
More informationHigher-order ordinary differential equations
Higher-order ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called
More informationProblem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv
V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =
More informationLesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.
Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an
More informationEngg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationMa 221 Final Exam Solutions 5/14/13
Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes
More informationMath 240 Calculus III
DE Higher Order Calculus III Summer 2015, Session II Tuesday, July 28, 2015 Agenda DE 1. of order n An example 2. constant-coefficient linear Introduction DE We now turn our attention to solving linear
More informationLecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem
Lecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem Math 392, section C September 14, 2016 392, section C Lect 5 September 14, 2016 1 / 22 Last Time: Fundamental Theorem for Line Integrals:
More informationSeries Solutions of Linear ODEs
Chapter 2 Series Solutions of Linear ODEs This Chapter is concerned with solutions of linear Ordinary Differential Equations (ODE). We will start by reviewing some basic concepts and solution methods for
More information17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
More informationReview for Ma 221 Final Exam
Review for Ma 22 Final Exam The Ma 22 Final Exam from December 995.a) Solve the initial value problem 2xcosy 3x2 y dx x 3 x 2 sin y y dy 0 y 0 2 The equation is first order, for which we have techniques
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS Basic concepts: Find y(x) where x is the independent and y the dependent varible, based on an equation involving x, y(x), y 0 (x),...e.g.: y 00 (x) = 1+y(x) y0 (x) 1+x or,
More informationPolytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012
Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet
More information4 Differential Equations
Advanced Calculus Chapter 4 Differential Equations 65 4 Differential Equations 4.1 Terminology Let U R n, and let y : U R. A differential equation in y is an equation involving y and its (partial) derivatives.
More information2.3 Linear Equations 69
2.3 Linear Equations 69 2.3 Linear Equations An equation y = fx,y) is called first-order linear or a linear equation provided it can be rewritten in the special form 1) y + px)y = rx) for some functions
More informationNATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS
MTH 421 NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS MTH 421 ORDINARY DIFFERENTIAL EQUATIONS COURSE WRITER Prof.
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More information5.4 Variation of Parameters
202 5.4 Variation of Parameters The method of variation of parameters applies to solve (1) a(x)y + b(x)y + c(x)y = f(x). Continuity of a, b, c and f is assumed, plus a(x) 0. The method is important because
More informationTitle: Solving Ordinary Differential Equations (ODE s)
... Mathematics Support Centre Title: Solving Ordinary Differential Equations (ODE s) Target: On completion of this workbook you should be able to recognise and apply the appropriate method for solving
More informationA Brief Review of Elementary Ordinary Differential Equations
A A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More informationModule 2: First-Order Partial Differential Equations
Module 2: First-Order Partial Differential Equations The mathematical formulations of many problems in science and engineering reduce to study of first-order PDEs. For instance, the study of first-order
More information0.1 Problems to solve
0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)
More informationFourier transforms. c n e inπx. f (x) = Write same thing in an equivalent form, using n = 1, f (x) = l π
Fourier transforms We can imagine our periodic function having periodicity taken to the limits ± In this case, the function f (x) is not necessarily periodic, but we can still use Fourier transforms (related
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 15 Method of Undetermined Coefficients
More informationFirst order Partial Differential equations
First order Partial Differential equations 0.1 Introduction Definition 0.1.1 A Partial Deferential equation is called linear if the dependent variable and all its derivatives have degree one and not multiple
More informationUNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH *
4.4 UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH 19 Discussion Problems 59. Two roots of a cubic auxiliary equation with real coeffi cients are m 1 1 and m i. What is the corresponding homogeneous
More informationFirst-Order ODE: Separable Equations, Exact Equations and Integrating Factor
First-Order ODE: Separable Equations, Exact Equations and Integrating Factor Department of Mathematics IIT Guwahati REMARK: In the last theorem of the previous lecture, you can change the open interval
More informationMathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited
Everything You Need to Know A Level Edexcel C4 March 4 Copyright 4 Elite Learning Limited Page of 4 Further Binomial Expansion: Make sure it starts with a e.g. for ( x) ( x ) then use ( + x) n + nx + n(n
More informationM343 Homework 3 Enrique Areyan May 17, 2013
M343 Homework 3 Enrique Areyan May 17, 013 Section.6 3. Consider the equation: (3x xy + )dx + (6y x + 3)dy = 0. Let M(x, y) = 3x xy + and N(x, y) = 6y x + 3. Since: y = x = N We can conclude that this
More informationIntroductory Differential Equations
Introductory Differential Equations Lecture Notes June 3, 208 Contents Introduction Terminology and Examples 2 Classification of Differential Equations 4 2 First Order ODEs 5 2 Separable ODEs 5 22 First
More informationNATIONAL ACADEMY DHARMAPURI TRB MATHEMATICS DIFFERENTAL EQUATIONS. Material Available with Question papers CONTACT ,
NATIONAL ACADEMY DHARMAPURI TRB MATHEMATICS DIFFERENTAL EQUATIONS Material Available with Question papers CONTACT 8486 17507, 70108 65319 TEST BACTH SATURDAY & SUNDAY NATIONAL ACADEMY DHARMAPURI http://www.trbtnpsc.com/013/07/trb-questions-and-stu-materials.html
More information1 Arithmetic calculations (calculator is not allowed)
1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem 1.4. 78 5 6 + 24 3 4 99 1
More informationA( x) B( x) C( x) y( x) 0, A( x) 0
3.1 Lexicon Revisited The nonhomogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x) F( x), A( x) dx dx The homogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x), A( x) dx
More informationكلية العلوم قسم الرياضيات المعادالت التفاضلية العادية
الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا
More informationBasic Theory of Linear Differential Equations
Basic Theory of Linear Differential Equations Picard-Lindelöf Existence-Uniqueness Vector nth Order Theorem Second Order Linear Theorem Higher Order Linear Theorem Homogeneous Structure Recipe for Constant-Coefficient
More informationChapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)
Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =
More informationBranch: Name of the Student: Unit I (Fourier Series) Fourier Series in the interval (0,2 l) Engineering Mathematics Material SUBJECT NAME
13 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE UPDATED ON : Transforms and Partial Differential Equation : MA11 : University Questions :SKMA13 : May June 13 Name of the Student: Branch: Unit
More informationChapter1. Ordinary Differential Equations
Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that
More informationOrdinary differential equations Notes for FYS3140
Ordinary differential equations Notes for FYS3140 Susanne Viefers, Dept of Physics, University of Oslo April 4, 2018 Abstract Ordinary differential equations show up in many places in physics, and these
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationElementary ODE Review
Elementary ODE Review First Order ODEs First Order Equations Ordinary differential equations of the fm y F(x, y) () are called first der dinary differential equations. There are a variety of techniques
More informationLECTURE NOTES OF DIFFERENTIAL EQUATIONS Nai-Sher Yeh
LECTURE NOTES OF DIFFERENTIAL EQUATIONS Nai-Sher Yeh June 2; 2009 Differential Equations 2 Introduction. Ordinary Differential Equation Def. A functional equation containing a function and its derivatives
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More informationAn Overly Simplified and Brief Review of Differential Equation Solution Methods. 1. Some Common Exact Solution Methods for Differential Equations
An Overly Simplified and Brief Review of Differential Equation Solution Methods We will be dealing with initial or boundary value problems. A typical initial value problem has the form y y 0 y(0) 1 A typical
More informationMATH 2250 Final Exam Solutions
MATH 225 Final Exam Solutions Tuesday, April 29, 28, 6: 8:PM Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the exam
More informationAdvanced Eng. Mathematics
Koya University Faculty of Engineering Petroleum Engineering Department Advanced Eng. Mathematics Lecture 6 Prepared by: Haval Hawez E-mail: haval.hawez@koyauniversity.org 1 Second Order Linear Ordinary
More informationLOWELL WEEKLY JOURNAL.
Y 5 ; ) : Y 3 7 22 2 F $ 7 2 F Q 3 q q 6 2 3 6 2 5 25 2 2 3 $2 25: 75 5 $6 Y q 7 Y Y # \ x Y : { Y Y Y : ( \ _ Y ( ( Y F [ F F ; x Y : ( : G ( ; ( ~ x F G Y ; \ Q ) ( F \ Q / F F \ Y () ( \ G Y ( ) \F
More informationODE classification. February 7, Nasser M. Abbasi. compiled on Wednesday February 07, 2018 at 11:18 PM
ODE classification Nasser M. Abbasi February 7, 2018 compiled on Wednesday February 07, 2018 at 11:18 PM 1 2 first order b(x)y + c(x)y = f(x) Integrating factor or separable (see detailed flow chart for
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationSecond-Order Linear ODEs
C0.tex /4/011 16: 3 Page 13 Chap. Second-Order Linear ODEs Chapter presents different types of second-order ODEs and the specific techniques on how to solve them. The methods are systematic, but it requires
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationProcedure used to solve equations of the form
Equations of the form a d 2 y dx 2 + bdy dx + cy = 0 (5) Procedure used to solve equations of the form a d 2 y dx 2 + b dy dx 1. rewrite the given differential equation + cy = 0 (1) a d 2 y dx 2 + b dy
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential
More informationLecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI
Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still
More informationPUTNAM PROBLEMS DIFFERENTIAL EQUATIONS. First Order Equations. p(x)dx)) = q(x) exp(
PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS First Order Equations 1. Linear y + p(x)y = q(x) Muliply through by the integrating factor exp( p(x)) to obtain (y exp( p(x))) = q(x) exp( p(x)). 2. Separation of
More informationOrdinary Differential Equations (ODEs)
Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationOrdinary Differential Equations Lake Ritter, Kennesaw State University
Ordinary Differential Equations Lake Ritter, Kennesaw State University 2017 MATH 2306: Ordinary Differential Equations Lake Ritter, Kennesaw State University This manuscript is a text-like version of the
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationGreen Lab. MAXIMA & ODE2. Cheng Ren, Lin. Department of Marine Engineering National Kaohsiung Marine University
Green Lab. 1/20 MAXIMA & ODE2 Cheng Ren, Lin Department of Marine Engineering National Kaohsiung Marine University email: crlin@mail.nkmu.edu.tw Objectives learn MAXIMA learn ODE2 2/20 ODE2 Method First
More informationTheory of Higher-Order Linear Differential Equations
Chapter 6 Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory A linear differential equation of order n has the form a n (x)y (n) (x) + a n 1 (x)y (n 1) (x) + + a 0 (x)y(x) = b(x), (6.1.1)
More information1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =
DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log
More informationSpecial Mathematics Tutorial 1
Special Mathematics Tutorial 1 February 018 ii Science is a differential equation. Relegion is a boundary condition Alan Turing 1 Differential equations A case for Sherlock Holmes London, 18.30 o clock.
More informationMATH 307: Problem Set #3 Solutions
: Problem Set #3 Solutions Due on: May 3, 2015 Problem 1 Autonomous Equations Recall that an equilibrium solution of an autonomous equation is called stable if solutions lying on both sides of it tend
More informationSummer 2017 MATH Solution to Exercise 5
Summer 07 MATH00 Solution to Exercise 5. Find the partial derivatives of the following functions: (a (xy 5z/( + x, (b x/ x + y, (c arctan y/x, (d log((t + 3 + ts, (e sin(xy z 3, (f x α, x = (x,, x n. (a
More informationMath 23: Differential Equations (Winter 2017) Midterm Exam Solutions
Math 3: Differential Equations (Winter 017) Midterm Exam Solutions 1. [0 points] or FALSE? You do not need to justify your answer. (a) [3 points] Critical points or equilibrium points for a first order
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS 1. Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt
More informationDifferential Equations Revision Notes
Differential Equations Revision Notes Brendan Arnold January 18, 2004 Abstract These quick refresher notes will probably only be useful for those with a univsersity level maths. They were written primarily
More informationMath 266, Midterm Exam 1
Math 266, Midterm Exam 1 February 19th 2016 Name: Ground Rules: 1. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use
More informationLINEAR DIFFERENTIAL EQUATIONS. Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose. y(x)
LINEAR DIFFERENTIAL EQUATIONS MINSEON SHIN 1. Existence and Uniqueness Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose p 1 (x),..., p n (x) and g(x) are continuous real-valued
More information